What symbols do you find in the story ''Barbie-Q''? Is there a commonality to them?

The story "Barbie-Q" reads almost like a prose poem because of the way it throws huge bursts of imagery at the reader, completely immersing them in the moment and putting them in the mind of the excited young girls. Every time a particular barbie ensemble is mentioned, the description of every piece of the outfit is frantically listed afterward. On a surface level, this is symbolic of the girls's obsession with the dolls themselves. On another level, it represents the performative expectation of women in the world. The story mentions several times that the two girls are from families of meager means; there is no doubt that they are already familiar with their perceived economic shortcomings. This leads to a moment of personal triumph when they receive dolls that survived a toy warehouse fire. Despite their wear and damage, the girls still love them.

The symbols in Cisneros's story are the Barbies—both those the characters have originally and those that come from the warehouse fire. The original Barbies are symbols of mid-century American glamor, as they include stiletto heels, an A-line coat dress that comes with a Jackie Kennedy–style pillbox hat, and a glittery black evening dress with formal gloves.
Later, the characters acquire smoke-damaged Barbies at a flea market after a warehouse has burned down. These toys are sooty and soaked with water, and one of the Barbies has a melted foot. These Barbies have a commonality with the more pristine Barbies, but they are secondhand toys that symbolize the breakdown of the American Dream. Poorer children like those in the story, whose families can't afford to buy their children endless Barbies and outfits, must make do with water-stained and smoke-damaged toys. Their Barbies are not pristine but have been victims of danger and poor treatment, just as they likely have been as well.

This Sandra Cisneros story revolves around the Barbie obsession of two young girls, highlighting their love of fashion and dolls as well as the fact that they're unable to afford new Barbies for their collections. The symbols used in the story have common elements of youthful innocence, societal expectations, poverty, and gratitude.
The tale begins with a description of outfits worn by the girls' dolls, including the iconic striped swimsuit of original Barbie, as well as their sole set of spare outfits, consisting of elegant evening-wear choices. The classic, formal styles the girls have chosen for their dolls represents an admiration for glamour, high society, and unattainable status symbols they may never be able to afford. The script the girls use when playing Barbie details a stereotypical love triangle between one man and two women and symbolizes the girls' desire to grow up, gain freedom, and make their own decisions. Insults such as "dumbbell" and "stinky" in the dolls' confrontation demonstrate the naiveté of two young girls excited by the glamour of growing up. The fire in the toy warehouse on Halsted Street symbolizes how quickly one's fortune can change, as they are suddenly presented with the opportunity to purchase slightly damaged Barbies at a low cost. The elation the girls feel at unexpectedly adding brand new, though imperfect, dolls to their collections symbolizes the innocence and resilience of youth and demonstrates how easily contentment can be found when one's expectations are grounded in reality.

Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 43

Applying Root test on a series sum a_n , we determine the limit as:
lim_(n-gtoo) root(n)(|a_n|)= L
or
lim_(n-gtoo) |a_n|^(1/n)= L
Then, we follow the conditions:
a) Llt1 then the series is absolutely convergent.
b) Lgt1 then the series is divergent.
c) L=1 or does not exist then the test is inconclusive. The series may be divergent, conditionally convergent, or absolutely convergent.
For the given series sum_(n=1)^oo(2root(n)(n)+1)^n , we have a_n =(2root(n)(n)+1)^n .
Applying the Root test, we set-up the limit as:
lim_(n-gtoo) |(2root(n)(n)+1)^n|^(1/n) =lim_(n-gtoo) ((2root(n)(n)+1)^n)^(1/n)
Apply the Law of Exponents: (x^n)^m= x^(n*m) .
lim_(n-gtoo) ((2root(n)(n)+1)^n)^(1/n) =lim_(n-gtoo) (2root(n)(n)+1)^(n*(1/n))
=lim_(n-gtoo) (2root(n)(n)+1)^(n/n )
=lim_(n-gtoo) (2root(n)(n)+1)^1
=lim_(n-gtoo) (2root(n)(n)+1)
Evaluate the limit.
lim_(n-gtoo) (2root(n)(n)+1) =lim_(n-gtoo) 2root(n)(n)+lim_(n-gtoo)1
=2lim_(n-gtoo) root(n)(n)+lim_(n-gtoo)1
= 2 * 1 + 1
=2 +1
=3
Note: lim_(n-gtoo) 1 =1 and lim_(n-gtoo) root(n)(n) =lim_(n-gtoo) n^(1/n) =1 .
The limit value L =3 satisfies the condition: Lgt1 .
Conclusion: The series sum_(n=1)^oo(2root(n)(n)+1)^n is divergent .

What is the best word to describe Alice's character in "How I Met My Husband"?

In the short story "How I Met My Husband," written by Alice Munro, the handsome pilot Chris Watters is already engaged to Alice Kelling when Edie meets and falls in love with him. 
When Edie first sees Alice, she describes her appearance negatively, thinking that there was "nothing in the least pretty or even young-looking about her." It's clear that Edie is being a bit petty and jealous when she says this, but it's also very clear that Chris is not as interested in Alice as Alice is in him. 
A good word to describe Alice would be unrelenting. We can tell that she has been following Chris around for a while, trying to keep him in one place, and she shows all spite and no empathy for Edie when she finds out that Chris, a grown man, "was intimate" with a girl as young as her. 

Compare and contrast "The Age of Great Dreams" by David Farber and "The Sixties Unplugged" by Gerard DeGroot.

Both books are similar in that they acknowledge the major events and social developments of the 1960s; however, they differ in thematic focus. While The Sixties Unplugged focuses on the theme of perception versus reality, The Age of Great Dreams focuses on causative factors that led to major societal shifts in the 1960s.
In the fifteen chapters of his book The Sixties Unplugged, DeGroot debunks many of the myths that surrounded the 1960s. He argues that the idea of the Sixties as an era of peace, tolerance, and new beginnings was more idealistic naivety than reality. DeGroot's 65 separate vignettes may prove disjointed, but they powerfully debunk the widely held myths associated with the era. On the other hand, Farber chooses to focus on how events in the 1950s led to major social developments during the 1960s. The differing focus in both books can be seen most clearly in the authors' approach towards the Vietnam War and the Civil Rights Movement.
In The Sixties Unplugged, the Vietnam War is a microcosmic representation of the social turbulence that encapsulated the 1960s; DeGroot uses the war as a basis for discussing Civil Rights issues, the anti-war movement, youthful discontent, economic instability, and the American music industry.
For example, DeGroot references Muhammad Ali's protest on being drafted. The famed pugilist eventually filed for CO (conscientious objector) status, arguing that he had been unfairly targeted because of his Muslim faith. DeGroot relates that Ali was exiled from boxing for three years for his refusal to fight in the Vietnam War. However, he also argues that Ali's plight brings to light two important issues: that the court ruling against Ali was harsh but indicative of the experiences of thousands who were forced to fight and that dismissing the justice system as corrupt demeaned the efforts of those whose conscientious objector status was genuine. Here, DeGroot chooses not to take sides but to focus on the conflicting arguments that surrounded the war draft.
Farber, on the other hand, devotes three chapters to discussing the Vietnam War. His first chapter on the subject, appropriately titled Vietnam, provides the reader with the causative factors that led to the Vietnam War. Farber relates that the Vietnam War actually began during WWII, when American OSS (Office of Strategic Services) agents joined forces with the Communist Vietminh against the Japanese. After Vietnam was liberated from Japanese control, President Truman began to modify his views regarding the South East Asian nation. In the aftermath of WWII, two powerful nations began to emerge as global power players: the United States and the USSR.
The United States began to fear that Russian hegemonic ambitions would deliver South East Asia to communist control. Farber argued that such a development would bode ill for American interests in the region. The United States had a vested interest in the resources and markets of the region, and Communist control would shut off its access to those regions. Essentially, in the 1950s, the United States began to view Ho Chih Minh's objective for an independent Vietnam as counterproductive to American economic and military interests.
Meanwhile, after the Geneva peace agreement in 1954, Vietnam was temporarily divided into two regions. The United States hoped that it could set up a South Vietnamese government (sympathetic to American interests) to challenge Ho Chih Minh's Communist North. The Cold War essentially began in the 1950s, with the United States on the side of South Vietnam and Russia (along with newly Communist China) on the side of the Viet Cong. Essentially, this 1950s power struggle between Russia and the United States propelled America into the Vietnam War in the 1960s. In A Nation At War, Farber discusses the anti-war sentiment fomenting across college campuses in light of the draft. Meanwhile, in the next chapter, aptly titled The War Within, Farber addresses how the Vietnam War birthed a national political divide that threatened the social stability of the United States in the late 60s and the 70s. Additionally, he discusses the emergence of a drug culture and a free love culture that descended on the United States in the aftermath of the Vietnamese War.
In his book, Farber also discusses the roots of the Civil Rights movement in the United States. He documents how the rise of moderate integrationist theories and the emergence of Black nationalism resulted in conflicts that haunted the Civil Rights Movement. 
As can be seen, both books differ in thematic focus. While DeGroot chooses to debunk the myths of the 1960s, Farber focuses on discussing the causative factors that fueled major societal changes in the 1960s.

dy/dx = 6x^2 Use integration to find a general solution to the differential equation

An ordinary differential equation (ODE)  is differential equation for the derivative of a function of one variable. When an ODE is in a form of y'=f(x,y) , this is just a first order ordinary differential equation. 
The y ' is the same as (dy)/(dx) therefor first order ODE can written in a form of (dy)/(dx) = f(x,y)
That is form of the given problem: (dy)/(dx) = 6x^2.
We may apply integration after we rearrange it in a form of variable separable differential equation: N(y) dy = M(x) dx .
By cross-multiplication, we can be rearrange the problem into: (dy) = 6x^2dx .
Apply direct integration on both sides:
int (dy) =int 6x^2dx .
For the left side, we may apply basic integration property: 
int (dy)=y
For the right side, we may apply the basic integration property: int c*f(x)dx = c int f(x) dx .
int 6x^2dx =6int x^2dx
 Then apply Power Rule for integration: int u^n du= u^(n+1)/(n+1)+C
6 int x^2dx = 6*x^(2+1)/(2+1)
                  = 6*x^3/3+C
                  = 2x^3+C
 
Combining the results, we get the general solution for differential equation:
y=2x^3+C
 

How is the image of a “boot stamping on a human face—forever” an appropriate image of this future anti-utopia in 1984?

In the dystopian nation of Oceania, the authoritarian regime oppresses the entire population through inhumane tactics and is dedicated to completely eradicating individuality. The Party's primary goal is to forever dominate and control the population of Oceania. The Party maintains a hysterical, threatening environment at all times, keeps the citizens under constant surveillance, publicly executes enemies of the state, and tortures political dissidents. The government also requires every Party member to worship Big Brother and completely accept their absurd propaganda. If Party members are not completely orthodox, they are tortured and brainwashed in the Ministry of Love.
In Emmanuel Goldstein's book, The Theory and Practice of Oligarchical Collectivism, Winston Smith reads about the various methods the government utilizes to maintain power and oppress the citizens. Winston reads about the necessity of continual warfare to use up valuable resources as well as the ideology of Ingsoc. Since the Party is dedicated to eradicating individuality and creating an oppressive, threatening environment, the image of a boot stomping on a human face is an appropriate image of the dystopian future under the Party's reign. The boot symbolically represents the tyrannical presence of the government and the squashed human face is the population of Oceania under Big Brother's authoritative rule. As Winston learns while he is being tortured by O'Brien, the Party's primary goal is to dominate the population and rule forever. Under the Party's rule, individuals will continue to suffer like the person whose face is being mercilessly smashed by the boot.

As has already been pointed out, the image in question is highly appropriate for the dystopia of 1984, but I would add that this image of an act of violence is additionally appropriate because of how it relates to the actual goals and motivations of the Inner Party itself. One of the key observations in 1984 is that, when you dig beneath the propaganda and the rhetoric, totalitarian states are at their core primarily about domination. We see this theme personified in O'Brien. O'Brien does not hold any ideology or justification for the things he's done, because for O'Brien, the means and the ends amount to the same thing. His position is about exercising power, for no other purpose than to wield that power over everyone else: it's literally oppression for oppression's own sake. That's the goal of the Party.
With this in mind, consider the image at the heart of your question. We're looking at an act of violence and of domination: one person stepping on the collective face of humanity. The key thing to keep in mind about the image, though, is that this image doesn't just apply to the tools of oppression that the State uses, it's also describing the motivation driving the State to act the way it does. The State oppresses just because it can.

The image of a boot stamping on the human face forever is an appropriate image of the anti-utopia established in Orwell's 1984.
The anti-utopia (or dystopia) of 1984 features citizens who are victims of an ongoing war, constant government surveillance, and blatant public manipulation. The government has even invented a language to replace traditional English, and the Thought Police control absolutely every facet of human life. There is no privacy or dignity for residents of this society. There is persecution for individualism and independent thinking; and the punishment for trying to break free from this oppression is horrific.
The image of a boot stamping on the human face forever is a perfect representation of the government oppressing its citizens and literally "stamping out" anything that makes them human (thoughts, feelings, individual expression, etc.).

Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 10

Arc length (L) of the function y=f(x) on the interval [a,b] is given by the formula,
L=int_a^bsqrt(1+(dy/dx)^2)dx , if y=f(x) and a <= x <= b ,
y=3/2x^(2/3)+4
Now let's differentiate the function with respect to x,
dy/dx=3/2(2/3)x^(2/3-1)
dy/dx=1/x^(1/3)
Plug in the above derivative in the arc length formula,
L=int_1^27sqrt(1+(1/x^(1/3))^2)dx
L=int_1^27sqrt(1+1/x^(2/3))dx
L=int_1^27sqrt((x^(2/3)+1)/x^(2/3))dx
L=int_1^27sqrt(x^(2/3)+1)/x^(1/3)dx
Now let's first evaluate the definite integral by using integral substitution,
Let u=x^(2/3)+1
(du)/dx=2/3x^(2/3-1)
(du)/dx=2/(3x^(1/3))
intsqrt(x^(2/3)+1)/x^(1/3)dx=intsqrt(u)3/2du
=3/2intsqrt(u)du
=3/2((u)^(1/2+1)/(1/2+1))
=3/2(u^(3/2)/(3/2))
=u^(3/2)
Substitute back u=x^(2/3)+1 and add a constant C to the solution,
=(x^(2/3)+1)^(3/2)+C
L=[(x^(2/3)+1)^(3/2)]_1^27
L=[(27^(2/3)+1)^(3/2)]-[(1^(2/3)+1)^(3/2)]
L=[(9+1)^(3/2)]-[2^(3/2)]
L=[10^(3/2)]-[2^(3/2)]
L=31.6227766-2.828427125
L=28.79434948
Arc length of the function over the given interval is ~~28.79435